Algebra retracts and Stanley-Reisner rings (Q2443278)

Let \(\vartheta: R\rightarrow S\) be an injective ring homomorphism, which is a ring extension that admits a ring homomorphism \(\varphi: S\rightarrow R\) such that \(\varphi\circ \vartheta=\text{Id}_R\). The ring \(R\) is said to be an algebra retract of \(S\). Analogously graded algebra retract is defined. \textit{W. Bruns} and \textit{J. Gubeladze} [Trans. Am. Math. Soc. 354, No. 1, 179--203 (2002; Zbl 1012.18005)] conjectured that every graded algebra retract of a polytopal algebra over a field \(\mathbb{K}\) is also a polytopal algebra over \(\mathbb{K}\). Motivated by this conjecture, which remains still open, the authors are proving in their main result that every graded algebra retract of a Stanley-Reisner ring is a Stanley-Reisner ring. In order to prove this, the authors are proving firstly, that every graded algebra retract of a Stanley-Reisner ring has a base. Also, the authors are proving by a homological way, that every algebra retract of a regular ring, is also regular. This descent of regularity along algebra retracts was first proved in [\textit{D. L. Costa}, J. Algebra 44, 492--512 (1977; Zbl 0352.13008)] by a non-homological method. Furthermore, the authors give an example of a Stanley-Reisner ring \(S\) which is Gorenstein and a Stanley-Reisner ring \(R\) which is Cohen-Macaulay such that \(R\) is an algebra retract of \(S\). Finally, the authors are moving on the general class of monomial quotient rings. They are proving that if \(I=(x_{i_1}^{d_{i_1}},\ldots,x_{i_s}^{d_{i_s}})\) is an irreducible monomial ideal of \(S\), then every graded algebra retract of \(S/I\) has a base.